Displacement measuring interferometers monitor changes in the position of a measurement object relative to a reference object based on an optical interference signal. The interferometer generates the optical interference signal by overlapping and interfering a measurement beam reflected from the measurement object with a reference beam reflected from the reference object.
In many applications, the measurement and reference beams have orthogonal polarizations and different frequencies. The different frequencies can be produced, for example, by laser Zeeman splitting, by acousto-optical modulation, or internal to the laser using birefringent elements or the like. The orthogonal polarizations allow a polarization beam-splitter to direct the measurement and reference beams to the measurement and reference objects, respectively, and combine the reflected measurement and reference beams to form overlapping exit measurement and reference beams. The overlapping exit beams form an output beam that subsequently passes through a polarizer. The polarizer mixes polarizations of the exit measurement and reference beams to form a mixed beam. Components of the exit measurement and reference beams in the mixed beam interfere with one another so that the intensity of the mixed beam varies with the relative phase of the exit measurement and reference beams.
A detector measures the time-dependent intensity of the mixed beam and generates an electrical interference signal proportional to that intensity. Because the measurement and reference beams have different frequencies, the electrical interference signal includes a “heterodyne” signal having a beat frequency equal to the difference between the frequencies of the exit measurement and reference beams. If the lengths of the measurement and reference paths are changing relative to one another, e.g., by translating a stage that includes the measurement object, the measured beat frequency includes a Doppler shift equal to 2νnp/λ, where ν is the relative speed of the measurement and reference objects, λ is the wavelength of the measurement and reference beams, n is the refractive index of the medium through which the light beams travel, e.g., air or vacuum, and p is the number of passes to the reference and measurement objects. Changes in the phase of the measured interference signal correspond to changes in the relative position of the measurement object, e.g., a change in phase of 2π corresponds substantially to a distance change L of λ/(2np). Distance 2L is a round-trip distance change or the change in distance to and from a stage that includes the measurement object. In other words, the phase Φ, ideally, is directly proportional to L, and can be expressed asΦ=2pkL cos2 θ  (1)
for a plane mirror interferometer, e.g., a high stability plane mirror interferometer, where
  k  =            2      ⁢      π      ⁢                          ⁢      n        λ  and θ is the orientation of the measurement object with respect to a nominal axis of the interferometer. This axis can be determined from the orientation of the measurement object where Φ is maximized. Where θ is small, Equation (1) can be approximated byΦ=2pkL(1−θ2)  (2)
Unfortunately, this equality is not always exact. Many interferometers include non-linear effects such as those known as “cyclic errors.” The cyclic errors can be expressed as contributions to the phase and/or the intensity of the measured interference signal and have a sinusoidal dependence on the change in for example optical path length 2pnL. In particular, a first order cyclic error in phase has for the example a sinusoidal dependence on (4πpnL)/λ and a second order cyclic error in phase has for the example a sinusoidal dependence on 2(4πpnL)/λ. Higher order cyclic errors can also be present as well as sub-harmonic cyclic errors and cyclic errors that have a sinusoidal dependence of other phase parameters of an interferometer system comprising detectors and signal processing electronics. Different techniques for quantifying such cyclic errors are described in commonly owned U.S. Pat. Nos. 6,137,574, 6,252,688, and 6,246,481 by Henry A. Hill.
There are in addition to the cyclic errors, non-cyclic non-linear errors or non-cyclic errors. One example of a source of a non-cyclic error is the diffraction of optical beams in the measurement paths of an interferometer. Non-cyclic error due to diffraction has been determined for example by analysis of the behavior of a system such as found in the work of J.-P. Monchalin, M. J. Kelly, J. E. Thomas, N. A. Kurnit, A. Szöke, F. Zernike, P. H. Lee, and A. Javan, “Accurate Laser Wavelength Measurement With A Precision Two-Beam Scanning Michelson Interferometer,” Applied Optics, 20(5), 736-757, 1981.
A second source of non-cyclic errors is the effect of “beam shearing” of optical beams across interferometer elements and the lateral shearing of reference and measurement beams one with respect to the other. Beam shears can be caused, for example, by a change in direction of propagation of the input beam to an interferometer or a change in orientation of the object mirror in a double pass plane mirror interferometer such as a differential plane mirror interferometer (DPMI) or a high stability plane mirror interferometer (HSPMI). A change in relative beam shear can also be introduced by a lateral displacement of a measurement object that includes a retroreflector. The relative difference in wavefronts can be a consequence of wavefront errors present in respective components of an input beam to the interferometer, i.e. a common mode type of wavefront error, and/or generated by imperfections in optical components of the interferometer, i.e. a differential mode type of wavefront error.
The non-cyclic non-linearities may also be present in an interferometer as a result of a relative change in the wavefronts of the reference and measurement beam components of an output beam at a detector, i.e. a differential mode type of wavefront error, with or without a change in relative beam shear of the reference and measurement beam components of the output beam at the detector. The differential mode type of wavefront errors can be generated by a change in relative beam shear of reference and measurement beam components at one or more measurement and reference beam path locations in the interferometer although there may be substantially no changes in relative shear between the reference and measurement beam components of the output beam of the interferometer at the detector. In this case, the effects of the change in relative shear of reference and measurement beam components at one or more measurement and/or reference beam path locations generates a differential mode type of change in the wavefronts of the reference and measurement beam components of the output beam irrespective of whether there is a change in relative shear of reference and measurement beam components of the output beam at the detector.